[tMcGrath]

I think the standard reference is probably Spivak's 'Calculus on Manifolds' but this never really did it for me.

If you have a background in physics then some combination of Nakahara's 'Geometry, Topology and Physics' and Baez and Muniain's 'Gauge Fields, Knots and Gravity' might be good (I haven't included relativity textbooks as I assume it you have a background in GR then you have enough differential geometry).

An unusual recommendation that I think is really nice is 'Stochastic Models, Information Theory and Lie Groups' by Chirikjian. It covers a few other topics mentioned in this thread and is really nice. It's _extremely_ concrete and spells out a lot of calculations in great detail. Plus, the connection to engineering applications is much more obvious.

[imh]

Chirikjian's book looks really cool! Its website says that in volume 1 "The author reviews stochastic processes and basic differential geometry in an accessible way for applied mathematicians, scientists, and engineers." And I can't tell if that means 'brief review because this is a prereq to the book' or if this is a good first take on it. Do you know which it is?